Exploiting Factor Autocorrelation to Improve Risk Adjusted Returns

Kevin Oversby†

22 February 2014

ABSTRACT

The Fama-French three factor model is ubiquitous in modern finance. Returns are modeled as

a linear combination of a market factor, a size factor and a book-to-market equity ratio (or

“value”) factor. The success of this approach, since its introduction in 1992, has resulted in

widespread adoption and a large body of related academic literature.

The risk factors exhibit serial correlation at a monthly timeframe. This property is strongest in

the value factor, perhaps due to its association with global funding liquidity risk.

Using thirty years of Fama-French portfolio data, I show that autocorrelation of the value factor

may be exploited to efficiently allocate capital into segments of the US stock market. The

strategy outperforms the underlying portfolios on an absolute and risk adjusted basis. Annual

returns are 5% greater than the components and Sharpe Ratio is increased by 86%.

The results are robust to different time periods and varying composition of underlying

portfolios. Finally, I show that implementation costs are much smaller than the excess return

and that the strategy is accessible to the individual investor.

†Freelance Analyst, email: koversby@gmail.com, tel: (1) 604 603 1747

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1. Introduction

The Fama French Three Factor Model describes the expected return (r) on an asset as a result

of its relationship to three factors: market risk (Mkt), size risk (SMB), and “value” risk (HML):

(1)

where rf is the risk-free return rate and the coefficients measure the exposure to each risk.

SMB, which stands for Small Minus Big, measures the excess return received by investing in

stocks of companies with relatively small market capitalization. This additional return is often

referred to as the “size premium.” SMB is computed as the average return for the smallest 30%

of stocks minus the average return of the largest 30% of stocks in that month.

HML, which is short for High Minus Low, measures the “value premium” provided to investors

for investing in companies with high book-to-market (B/M) values. HML is computed as the

average return for the 50% of stocks with the highest B/M ratio minus the average return of the

50% of stocks with the lowest B/M ratio each month.

Since its introduction by Fama and French (1992), a vast literature has been published on all

facets of the model from correlation with global economic factors (Asness, Moskowitz and

Pedersen 2013) to practical applications to fund management (Doskov, Pekkala and Ribeiro

2013).

All three factors are time varying. This raises the following questions:

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1. Is it possible to reliably allocate capital to the factor(s) likely to outperform in the next

period?

2. Do the costs of implementing such a strategy exceed the benefit?

3. Is this type of strategy accessible to the individual investor?

The focus of this paper will be on the application of one specific property of the factors. That

property is the monthly serial correlation, or autocorrelation, of the factors as illustrated in

table 1:

Table 1: Factor serial correlation (lag 1 month)

Autocorrelation can be exploited to predict which segment of the market is likely to outperform

in the next month. Historically, the HML factor exhibits the most stable autocorrelation and is

therefore the primary focus of this paper. Kothari and Shanken (1998) note that if the B/M

effect is related to risk and liquidity it can reasonably be expected to persist to some degree in

the future. Asness, Moskowitz and Pedersen (2013) document liquidity effects in value and

momentum returns. They find a positive relationship between value and liquidity risk.

Kothari and Shanken find that book-to-market effect is mainly concentrated in small firms. A

paper by Loughran (1996) mirrors this finding for large firms. For the largest size quintile, which

Mkt-RF SMB HML

1975-1985 0.05 0.12 0.19

1985-1995 0.05 0.10 0.26

1995-2005 0.05 0.25 0.25

2005-present 0.20 -0.06 0.34

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accounts for about three-quarters of all market value, he concludes that B/M has no reliable

predictive power for returns in the 1963-94 period.

For these reasons, this paper studies return data from portfolios with firm size below the NYSE

median.

The rest of this paper is organized as follows. Section 2 describes data and methodology.

Section 3 presents the empirical findings. Section 4 conducts the robustness checks and Section

5 concludes.

2. Data and Methodology

All raw data in this paper are taken from the on-line library provided by Kenneth French1. The

portfolios are selected from the 6 portfolios formed on size and book-to-market. The precise

definition, taken from the website, is as follows:

“The portfolios, which are constructed at the end of each June, are the intersections of 2

portfolios formed on size (market equity, ME) and 3 portfolios formed on the ratio of

book equity to market equity (BE/ME). The size breakpoint for year t is the median NYSE

market equity at the end of June of year t. BE/ME for June of year t is the book equity

for the last fiscal year end in t-1 divided by ME for December of t-1. The BE/ME

breakpoints are the 30th and 70th NYSE percentiles.

1. http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

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The portfolios for July of year t to June of t+1 include all NYSE, AMEX, and NASDAQ

stocks for which we have market equity data for December of t-1 and June of t, and

(positive) book equity data for t-1. “

I use 30 years of monthly total-return data in my analyses. Portfolios are value weighted. For

the robustness checks in Section 4, equal weighted portfolios and 60 year histories are also

used.

2.1 Methodology

A realistic strategy must use only information that was available to the investor in real-time.

The strategy described below uses the autocorrelation property of the value factor, HML. This

property was strongly evident since at least 1975 (table 1) therefore this criterion is satisfied.

One discussion point, which applies to many of these types of study, is that of trading costs.

For more than a decade, suitable low-cost factor based funds have been available. However, in

the 1980s and earlier, trading costs were higher and use of appropriate funds may not have

been possible in lieu of individual stocks. The effect may be to reduce the level of realism as

test length increases.

The HML factor is time varying and autocorrelated, therefore the sign of the factor is a good

predictor of the following month’s sign. A positive (negative) HML factor in the current month

predicts value will outperform (underperform) growth next month. A strategy is proposed

which switches 100% of capital monthly from value (high B/M) to growth (low B/M) based on

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the sign of the HML factor. This switching portfolio is itself autocorrelated therefore the

strategy switches to risk-free if the previous monthly return is negative:

Let R be the monthly return with subscripts: p portfolio, v value, g growth, f risk-free and t time:

Rp,t = Rv,t if HMLt-1 > 0 and Rp,t-1 > 0

Rp,t = Rg,t if HMLt-1 < 0 and Rp,t-1 > 0

Rp,t = Rf,t if Rp,t-1 < 0

The value portfolio is the small high book-to-market Fama-French portfolio and the growth

portfolio is the small low book-to-market Fama-French portfolio from the 2x3 series.

Many tactical portfolios use bonds as a non-correlated asset to reduce return volatility (e.g.

Faber 2007). I choose to switch to risk-free rather than bonds to avoid biasing the results.

Bonds have been in a bull market over the duration of the tests and typically negatively

correlated with stocks. This return enhancer may not be available in the intermediate future,

given that interest rates are currently close to zero.

3. Results

Table 2 lists the results and statistics of the strategy compared with the individual components.

Sharpe Ratios are calculated relative to zero and annualized. I report the statistical significance

of the null hypothesis of zero mean return (t-stat). Results are frictionless; costs are discussed

later.

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Table 2: Results of the value-growth switching strategy

SR Sharpe Ratio, CAGR Compound Annual Growth Rate, VAL Value, GRO Growth

The strategy annual return is 3.4% greater than the average of the base Fama-French portfolios.

Sharpe Ratio increases from 0.7 to 1.1, signifying improved returns on a risk-adjusted basis.

3.1 Enhanced results

Can this strategy be improved without adding complexity? Kothari and Shanken (1998) show

that the growth portfolio has relatively low returns. Berger, Israel and Moskovitz (2009)

describe an alternative and improved way to access growth: momentum. This anomaly is

extremely robust and persists over most asset classes and time periods (Asness, Moskowitz and

Pedersen 2013). They note that momentum is closely correlated with growth but with 3.3%

greater annual returns from 1980-2009. The momentum portfolio has better performance,

both in absolute terms and relative to a core index.

Israel and Moskowitz (2013) show that the momentum premium is present and stable across all

size groups—there is little evidence that momentum is substantially stronger among small cap

stocks over the entire 86-year U.S. sample period. The value premium, on the other hand, is

largely concentrated only among small stocks. For consistency with the value portfolio, the

small cap momentum portfolio is used.

1984-2014 VAL GRO switch

SR 0.7 0.7 1.1

t-stat 3.7 3.8 5.9

CAGR 13.4% 10.6% 15.3%

R296.0% 87.3% 99.5%

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Replacing the growth portfolio with a momentum portfolio leads to the result in table 3.

Table 3: Results of the value-momentum switching strategy

SR Sharpe Ratio, CAGR Compound Annual Growth Rate, VAL Value, MOM Momentum

The annual return of 19.7% is 5.0% greater than the average of the component portfolios. Risk

adjusted returns (measured by the Sharpe Ratio) increase by 86% over the value portfolio with

a t-statistic of 7.1 and R2 of 99.7%.

The maximum drawdown (DD) is dramatically reduced from 62.8% to 18.3%.

Figure 1 shows compounded growth of $1 starting in 1984, for the value and momentum

portfolios, and the switching strategy. The final worth of the strategy portfolio is about $200.

The lower part of the plot depicts the state of the strategy: up = value, down = momentum and

center = risk-free. A moving average is overlaid for visualization of the market regime. The

strategy typically switches 7 times per year.

Notice how the strategy moves aggressively into value after the recessions of 2002 and 2008.

Daniel and Moskowitz (2013) find that momentum portfolios strongly underperform during this

type of recovery or ‘rebound’ period.

1984-2014 VAL MOM switch

SR 0.7 0.8 1.3

t-stat 3.7 4.4 7.1

CAGR 13.4% 16.1% 19.7%

R2

96.0% 95.8% 99.7%

Max. DD 62.8% 54.1% 18.3%

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Figure 1: Growth of $1 for the value-momentum switching strategy

3.2 Factor analysis

The portfolio returns were regressed on the risk factors using equation (1). Results are shown

in table 4.

Table 4: Portfolio factor analysis

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All portfolios load heavily on size, as expected due to the small portfolios used. Loading on the

market factor, also known as beta, is slightly above one for the base portfolios. Beta is much

lower for the switching strategy but the regression fit (R2) is poor. Adding the momentum

factor (UMD) does not improve R2 therefore those results are omitted.

3.3 Implementation costs

Annual fees

Small value funds such as VBR1 are available for 10 basis points (bp) annually. Small

momentum funds (for example DWAS2) are a more recent innovation and cost in the range of

60 bp. Assuming equal time in each fund, annual fees could average 35 bp.

Commissions and slippage

With today’s fixed brokerage costs, use of liquid funds and generous allowance for slippage,

trades could be completed for 5 bp. Annual trading costs for 7 round-trip trades total 70 bp.

Therefore the total implementation costs, excluding taxes, are 105 bp annually. This is a fifth of

the strategy excess return over the underlying component average.

4. Robustness checks

To check robustness, I repeated the calculations for the equally-weighted small value and

momentum portfolios. The results are shown in table 5.

1. Vanguard Small-Cap Value ETF on NYSE

2. PowerShares DWA Small Cap Momentum Portfolio on NASDAQ

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Table 5: Value-momentum strategy with equal-weight (ew) portfolios

SR Sharpe Ratio, CAGR Compound Annual Growth Rate, VAL Value, MOM Momentum

Performance is higher across the board for equal-weight portfolios (but may not be practically

realizable due to large positions in illiquid stocks). However, for the purpose of comparison, the

results show that the strategy generates a 78% improvement in Sharpe Ratio and an annual

return increase of 5.6% over the base portfolio average.

Finally, the calculations were repeated for the time period from 1954 to 1984. Tests are

frictionless which, as alluded to in the methodology section, would be less realistic for this

epoch. The results are shown in table 6.

Table 6: Value-momentum strategy from 1954 to 1984

SR Sharpe Ratio, CAGR Compound Annual Growth Rate, VAL Value, MOM Momentum

The strategy underperforms momentum by 1.3% annually on an absolute basis but the Sharpe

Ratio of the strategy is 50% greater than the value portfolio and 36% superior to the

1984-2014 VAL(ew) MOM(ew) switch

SR 0.9 0.9 1.6

t-stat 5.0 4.9 8.7

CAGR 17.3% 18.0% 23.2%

R296.7% 96.7% 99.5%

1954-1984 VAL MOM switch

SR 1.0 1.1 1.5

t-stat 5.4 5.9 8.1

CAGR 18.6% 21.8% 20.5%

R296.4% 96.1% 96.2%

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momentum portfolio. Thus the risk adjusted returns are substantially larger. The strategy t-

statistic is higher than the base portfolios and R2 is similar throughout.

5. Concluding Remarks

The analyses in this paper show that the monthly autocorrelation property of the HML (or

“value”) factor can be reliably exploited to form an investment strategy. A strategy which

switches capital from value to momentum portfolios based on the sign of the factor is

demonstrated to have superior absolute and risk adjusted performance to the underlying

instruments.

The questions posed in the introduction have been answered as follows:

1. It is possible to reliably allocate capital using factor autocorrelation.

2. The costs of implementing such a strategy are a fifth of the excess returns.

3. The strategy uses liquid, low-cost funds and trades 7 times per year, therefore is readily

implementable by the individual investor.

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6. Bibliography

Antonacci, G., 2012, Risk Premia Harvesting Through Dual Momentum

Asness, C. S., T. J. Moskowitz, and L.H. Pedersen, 2013, Value and momentum everywhere, The

Journal of Finance 58, 929-895.

Berger A.L., Israel R. and Moskovitz T.J., 2009, The Case for Momentum Investing, AQR White

Paper

Daniel K.D. and Moskowitz T.J., 2013, Momentum Crashes, Working Paper

Doskov, N., Pekkala, T., Ribeiro, R., 2013, Tradable Aggregate Risk Factors and the Cross-

Section of Stock Returns.

Faber, M., 2007, A Quantitative Approach to Tactical Asset Allocation, Journal of Wealth

Management, Spring 2007

Fama, E. F., French, K. R., 1992, The Cross-Section of Expected Stock Returns, Journal of Finance

47, pp. 427–465.

Fama, E. F., French, K. R., 1993, Common risk factors in the returns on stocks and bonds, Journal

of Financial Economics 33, 3–56.

Fama, E. F., French, K. R., 2012, Size, Value, and Momentum in International Stock Returns

Israel R. and T.J. Moskowitz, 2013, The role of shorting, firm size, and time on market

anomalies, Journal of Financial Economics 108, 275-301.

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Kothari and Shanken, 1998, Beta and Book-to-Market: Is the Glass Half Full or Half Empty?

Loughran, T., 1997. Book-to-market across firm size, exchange, and seasonality: Is there an

effect? Journal of Financial and Quantitative Analysis 32, 249–268.

Sharpe, W. F., 1964, Capital asset prices: A theory of market equilibrium under conditions of

risk, Journal of Finance 19, pp. 425–442.